FEASIBILITY-BASED DESIGN MODEL FOR ROAD VERTICAL ALIGNMENT

. Road vertical alignment design is a multi-objective design problem that needs to consider multiple constraints. Intelligent design based on optimization algorithms cannot wholly solve problems, such as multi-objective, uncertainty, and constraint dynamics. The article proposes a model of dynamically transforming design constraints into feasible regions as the design develops, to provide decision information before design actions rather than performing constraint evaluation after the design that reduces the empirical estimation. The design actions are divided into new design actions and modifying design actions, and corresponding feasible regions derived from constraints of design specifications and control elevations are established, respectively. Geometrical equations and program algorithms of feasible regions are described in the graphic environment, which is applied to the vertical alignment design to improve the design efficiency and decision-making level.


Introduction
The traditional road vertical alignment design uses a heuristic design method supported by the CAD system. Based on their own design experience, the designer tentatively proposes a design solution, evaluates whether the design solution meets the design constraints, and then revises the design to complete the design task in repeated iterations. In this process, the role of the CAD system is for geometric calculations and the drawing of design solutions (Jha, Schonfeld & Jong, 2006;M. Singh, P. Singh & P. Singh, 2019).
Traditional design methods describe design problems based on qualitative descriptions of design experience. The design process needs to iterate repeatedly, which affects the design efficiency. Studies such as Jha and Schonfeld (2000), Beiranvand et al. (2017), Li et al. (2017) indicated that using optimization models could considerably speed up the design process and provide preliminary design solutions for alignment design. From the 1960s to the present, the research on vertical alignment design has mainly focused on the study of alignment optimization models.
Vertical alignment optimization is a process of finding the optimal solution through an optimization algorithm based on design specifications and constraints, intending to minimize road construction cost. Easa (1988), Moreb (1996) proposed a vertical alignment optimization model using linear programming with the goal of minimization of earthworks. A mixed-integer linear programming model considering blocks and side-slopes accurately estimates the optimal earthwork costs and reduces the calculation time (Hare, Lucet & Rahman, 2015;Hare, Koch & Lucet, 2011). Recently the genetic algorithms (GAs) have been used to solve vertical alignment optimization (Goktepe, Lav & Altun, 2009;Jha, Schonfeld & Jong, 2006). Enumeration, dynamic programming, and numerical search have been used to solve the vertical alignment optimization problem, which is summarised in detail by Jha, Schonfeld and Jong (2006). The discrete dynamic programming was integrated with the Weighted Ground Line Method (WGLM) to achieve the best vertical alignment of highways in terms of earthwork optimization (Goktepe, Altun & Ahmedzade, 2010). The alignment optimization model mathematically solves the problem of the optimal solution under certain constraints, but it lacks application in practical design. Hare et al. (2014) indicated that their optimization model was used by their industry partner and was developed as interactive road design software. Most of the papers did not mention software development and application.

Design process model based on feasible regions
Just as a design is a product of the designer's approach, it is also a reflection of a particular pattern of constraints that make up the problem (Lawson, 2006). The design of vertical alignment can be described as the problem of selecting grades and vertical curves and combining them reasonably under the constraints of design specifications and external environment to meet the design goals, such as function, safety, economy, and aesthetics. Design constraints are reflections of design goals.
In the process of design, from the initial proposal to the final design that meets the design goals, there are many intermediate design states.
The formation process of the intermediate design state is functionally represented as a series of design activities. The follow-up activities are the development and evolution of the previous activity, forming a design process composed of multiple design activities. In this process, design constraints are a mapping of the space of the design goals. Multiple design constraints tailor the design space to form a feasible region of the design feature space. The feasible region can be defined as follows: Definition 1. Feasible region: It refers to a design space which can meet various design constraints (including design specification constraints and elevation control constraints) in the process of vertical alignment design. As long as the VPI is set in this design space, the design solution will be feasible. The design process model M can be described as follows: where I / O is the input and output set; P is the design process set, Design states that satisfy certain constraints are transformed by mapping, which is the design sub-processes: The design input gets the design output through the design process: For the vertical alignment design process model (VADPM), the design process model input includes horizontal alignment, ground line of the alignment, etc. Design sub-processes P i refers to the design of the grade line and vertical curve, modification of vertical points of intersection (VPI), and vertical curve (VC). Design constraint set includes controls of design criteria, controls of elevation.
The vertical alignment design is generally developed gradually from the starting point to the endpoint. Design state of vertical alignment: where Stn i -the station of VPI i ; Elv i -the design elevation of VPI i ; K i -the rate of vertical curvature at VPI i . Satisfying design constraints is one of the core issues of the model. In interaction design, the designer, with the support of the CAD software, obtains a specific design state through a series of design actions. The software built-in Design Check module and the designer's experience determine whether the design state satisfies design constraints. If it is not satisfied, the design should be modified, and this is a designevaluation-modification iterative design method. In this method, the requirements of the design constraint C i+1 are met through empirical estimation of constraints in the design process P i+1 and design evaluation after reaching the design state Q i+1 . For P i+1 and Q i+1 , the design constraint C i+1 quantitatively acts after the completion of the stage design. Figure 1 shows that the design states Q i and Q i+1 are inherited.
During the development of Q i to Q i+1 , the design space under constraint tailoring constitutes the feasible region determined by P i+1 is within the feasible region, the design constraints C i+1 are satisfied. The feasible region model transforms the evaluation and modification after the design process into the evaluation and design under the limitation of the feasible region during the design process. Figure 2 shows that when the transition takes place from the design state Q i to Q i+1 , i.e., when determining the coordinate of VPI i+1 , the grade constraints include (AASHTO, 2011;Fanning et al., 2016;Wang, Guo & Luo, 2016): 1. For the reason of drainage, the minimum gradient should generally be higher than 0.3%, so the line l 1 in Fig. 2a is drawn; 2. Considering safety, economy, and efficiency, the maximum gradient should be controlled and the line l 2 in Fig. 2b should be drawn.
The line l 1 and l 2 form the feasible region in Fig. 2a. During the design, as long as VPI i+1 is guaranteed in the feasible region, grade control should be satisfied. Similarly, the feasible region is superimposed on the minimum grade length (l 3 ) and the critical length of grade (l 4 ), then the new feasible region that satisfies the constraints of the gradient and length of grade is formed. This is the idea of the design process model (FRDPM) based on the feasible region.
where FD -the design feasible region set; f -mapping of design constraints set to design feasible region set, where 1 , which means that the mapping from the design state Q i to Q i+1 is done by design process elements within the FD i+1 .
The vertical alignment design flow based on FRDPM is shown in Fig. 3. It can be seen that FRDPM transforms the constraint evaluation in the design process into a design within the feasible region. Compared with the traditional vertical alignment design approach, FRDPM could provide real-time graphical constraint information before design actions rather than constraint evaluation after design, which will reduce the work of repeated modifications and improve design efficiency.

Design constraints of vertical alignment
There are many constraints on vertical alignment design. This paper mainly discusses design specification constraints and control elevation constraints.
The vertical alignment comprises grades and the vertical curves between them, and design specification constraints include grade constraints and vertical curve constrains.
where C S -design specification constraints; C g -grade constraints; C c -vertical curve constrains.
where G -gradient between two VPI, %; G max -maximum gradient for specified design speed, %; G min -minimum gradient according to design specification, %; L -length of grade between two VPI, m; L min -minimum length of grade between two VPI, m; L max (G i ) -critical length of steep grade between two VPI for specified grade and design speed. The paper calculates the feasible region based on the minimum and maximum grade lengths in Chinese specifications (Wang, Guo & Luo, 2016).
where K -rate of vertical curvature; K min -minimum rate of the vertical curve for specified design speed; L c -length of the vertical curve, m; L cmin -minimum length of the vertical curve for specified design speed. Control elevation constraints include the following: control elevations for overpasses or underpasses to ensure vertical clearances; control elevations for bridges, culverts, or subgrades that consider flood levels; control elevations for tunnels based on geological conditions; and control elevations for intersection considering the intersecting road (Fwa, Chan & Sim, 2002;Kim et al., 2007). These constraints generate many tie points, defined as levels above which, through which or below which the profile must pass (Wolhuter, 2019).
where C E -control elevation constraints; C a -constraints of tie points above which the vertical alignment must pass; C b -constraints of tie points below which the vertical alignment must pass; C t -constraints of tie points through which the vertical alignment must pass.

VPI feasible region model for new design
Vertical alignment design can be divided into creating design objects and modifying design objects according to design actions. The present paper discusses the feasible region model for the new design and the feasible region model for design modifications.
3.1. Feasible region of C g Equation (7) shows that C g is composed of gradient constraint C g1 and grade length constraint C g2 : (10) Therefore, where FD A1 -feasible region of adding a new VPI under the constraint of C g ; FD g1 -feasible region of adding a new VPI under the constraint of C g1 ; FD g2 -feasible region of adding a new VPI under the constraint of C g2 .
Definition 3. Function of the critical length of steep grade: Taking the Chinese specification as an example, when the design speed is 80 km/h, the critical lengths of steep grade are shown in Table  1.   (  Fig. 5.

Feasible region of C g and C c
Once the VPI i+1 falls within the feasible region FD A1 , the design constraints on C g will be satisfied. However, whether vertical curves that satisfy design constraints can be set is still unknown. Therefore, the influence of C c on feasible regions should be considered. Vertical curves take the form of symmetrical parabolas, so the vertical curves have equal lengths on either side of VPI and can be designed using Eq. (15) (Higuera de Frutos & Castro, 2017).
where A i is the algebraic difference of gradient.
where G i+1 , G i are tangent grades in percent. As shown in Fig. 6 as long as VPI i+1 is not the EP (End Point), the necessary constraint for L i : Meanwhile, the necessary constraint for VC i : As shown in Eqs. (17) and (19) that can be regarded as the potential constraint of C c on VPI i+1 , the following equation can be obtained by mapping rules:

Feasible region of C S and C E
Once there are tie points, not only grade constraints C g and vertical curve constraints C c , but also control elevation constraints C E should be considered when adding a new VPI. Limited to space, only C a in C E is discussed. As shown in Fig. 9, point E is a tie point, and the vertical alignment should pass over it.
According to the coordinate of VPI i and point E, the feasible region FD A2 can be divided into three parts, as shown in Fig. 10.
The significance of each region: FD 1 : If VPI i is in this region, the constraint C a can be satisfied. FD 2 : If VPI i is in this region, the constraint C a may be satisfied. FD 3 : If VPI i is in this region, the constraint C a cannot be satisfied.
FD 3 is the impossible region and should be deleted. FD 1 and FD 2 should be retained and set in different styles for users' reference. Thus, can be defined as follows:  , .  FD A3 is shown in Fig. 11. Then the algorithm of determining VPI feasible region for adding a new VPI can be summarised as Algorithm 1 in the APPENDIX.

VPI feasible region model for design modifications
The modification of VPI includes inserting, deleting, and moving the VPI. The paper only discusses moving a middle VPI when the front VPI and the next VPI are fixed. As shown in Fig. 12, VPI i-1 and VPI i+1 are fixed. The movement of VPI i causes the front gradient, the rear gradient, and vertical curves to change. In this process, there is also a feasible region for VPI i .

Feasible region of C g
As shown in Fig. 12, when moving a middle VPI, its constraints mainly come from two parts: the front design constraint C f and the rear design constraint C r . Then where FD M1 -feasible region of moving a middle VPI under the constraint of C g ; FD f -feasible region of moving a middle VPI under the constraint of C f ; FD r -feasible region of moving a middle VPI under the constraint of C r .
The situation is the same for FD f and FD A1 , and FD r can be regarded as FD A1 in the opposite direction. Therefore, FD f and FD r are presented in Fig. 13, and FD M1 is shown in Fig. 14. 4.2. Feasible region of C g and C c As shown in Fig. 13, moving a middle VPI will also be constrained by three vertical curves: VC i-1 , VC i , and VC i+1 . The following equations can be obtained by mapping rules: Similarly, the situation is the same for FD cf and FD c1 , and FD cr can also be regarded as FD c1 in the opposite direction. Therefore, FD cf and FD cr are presented in Fig. 15 and Fig. 16, respectively.
Both the front and rear gradients are not constant, so FD cm is different from FD cf and FD cr . The constraint on VC i can be expressed by Eq. (25) FD cr Similar to FD c1 , FD cm can be defined as follows: (1) ;Stn FD cm is shown in Fig. 17. From Eq. (23), FD c2 is presented in Fig. 18. Similarly, FD M2 is presented in Fig. 19 based on Eq. (24).  Figure 19. Feasible region of FD M2

Feasible region of C S and C E
As shown in Fig. 20, if there is a tie point, control elevation constraints C E should also be considered when moving the VPI i . Limited to space, only C a in C E is discussed, which means the vertical alignment should pass over point E.
Similar to FD A2 , FD M2 can be divided into two parts, as shown in Fig. 21.
The significance of each region: FD 4 : If VPI i is in this region, the constraint C a can be satisfied.   ( D M3 is shown in Fig. 22. Then the algorithm of determining VPI feasible region for moving a middle VPI can be summarised as Algorithm 2 in the APPENDIX.

Conclusion
The paper has proposed a design process model based on feasible regions to provide decision information support for vertical alignment design. The main idea of the design process model is to convert constraints into feasible regions and provide designers with information support before design actions instead of design evaluation after design actions.
We discuss design specification constraints and control elevation constraints in vertical alignment design. Design specification constraints include grade, grade length, K-value, and length of vertical curve constraints. Control elevation constraints include elevation constraints for bridges, tunnels, intersections, overpasses, and underpasses based on flood levels or vertical clearances. We convert the above constraints into feasible region models for new and modified designs, respectively. Using mathematical and graphical methods, we give graphical numerical solutions and algorithm descriptions of feasible regions. Therefore, the feasible regions of the interactive design process are dynamically drawn to satisfy constraint information needs.
In addition to design specifications and control elevations, earthwork is a critical consideration in vertical alignment design. The design indices corresponding to each point in the feasible region is different, and the corresponding earthwork is also different. Combined with the cross-section design and earthwork calculation, the earthwork at each point in the feasible region can be theoretically calculated. Therefore, in the interactive design process, the earthwork in the feasible region can provide the designer with decision support for the cost goal, which is one of the future works.
Selecting reasonable points within the feasible region to determine the grade and vertical curve still requires some empirical judgment. There is a need for in-depth research on local optimization algorithms based on feasible regions, and dynamic optimization calculations in the interactive design process to improve design efficiency. C -Constraint set; C a -Constraint of tie points above which the vertical alignment must pass; C b -Constraint of tie points below which the vertical alignment must pass; C c -Vertical curve constraint; C E -Control-elevation constraint; C t -Constraint of tie points through which the vertical alignment must pass; C g -Grade constraint; C g1 -Gradient constraint; C g2 -Grade-length constraint; C S -Design-specification constraint; Elv i -Design elevation of VPI i ; f -Mapping of design-constraint set to design the feasible-region set; FD -Design feasible-region set; FD 1 -Definite feasible region derived from FD A2 ; FD 2 -Possible region derived from FD A2 ; FD 3 -Impossible region derived from FD A2 ; FD 4 -Definite feasible region derived from FD M2 ; FD 5 -Impossible region derived from FD M2 ; FD A1 -Feasible region of adding a new VPI under constraint C g ; FD A2 -Feasible region of adding a new VPI under constraints C g and C c ; FD A3 -Feasible region of adding a new VPI under constraints C S and C E ; FD c1 -Feasible region of adding a new VPI under constraint C c ; FD c2 -Feasible region of moving a middle VPI under constraint C c ; FD cf -Feasible region of moving a middle VPI under constraint VC i-1 ; FD cm -Feasible region of moving a middle VPI under constraint VC i ; FD cr -Feasible region of moving a middle VPI under constraint VC i+1 ; FD f -Feasible region of moving a middle VPI under constraint C f ; FD g1 -Feasible region of adding a new VPI under constraint C g1 ; FD g2 -Feasible region of adding a new VPI under constraint C g2 ; FD M1 -Feasible region of moving a middle VPI under constraint C g ; FD M2 -Feasible region of moving a middle VPI under constraints C g and C c ; FD M3 -Feasible region of moving a middle VPI under constraints C S and C E ; FD r -Feasible region of moving a middle VPI under constraint C r ; G -Gradient between two VPIs; G i -Gradient between VPI i-1 and VPI i ; G max -Maximum gradient according to the design specification; G min -Minimum gradient according to the design specification; I -Input set; K -Rate of vertical curvature; K i -Rate of vertical curvature at VPI i ; K imin -Minimum rate of vertical curvature at VPI i ; l 1 -Minimum longitudinal gradient constraint line;